\section{Problem 1}
			
	\subsection*{Question e}
			
			The real value of $r$ for the total number of records per vertex is given by the following formula :\\
		\begin{equation}
			r = \sqrt[d_{min}]{\dfrac{numberOfRecords + 1}{m + 1}} - 1
			\label{eq:avgrecordsmin}
		\end{equation}
	Our mistake was that $m$ must be set to $r$ value, and not as the biggest value it could be.
	
	This first formula was derived from this general one :
		\begin{equation}
			numberOfRecords = (m + 1)\times(r + 1)^{d} - 1
		\end{equation}
which leads to :
		\begin{displaymath}
			\begin{array}{r c l}
				numberOfRecords &=& (r + 1)\times(r + 1)^{d} - 1\\
				numberOfRecords &=& (r + 1)^{d + 1} - 1\\
				r &=& \sqrt[d + 1]{numberOfRecords + 1} -1
			\end{array}
		\end{displaymath}
			
		Numerical application :\\
		\begin{displaymath}
			r = \sqrt[8]{3\times10^{10} + 1} -1 = 19,400469236
		\end{displaymath}
			
			
		\fbox{\textbf{Answer :}	
			
		\begin{tabular}{|l|r|}
			\hline
				Maximum depth of the tree, $d_{min}$ & 7\\
			\hline
				Number of records in the root vertex, $m$ & 19,40\\
			\hline
				Average number of record in other vertices, $r$ & 19,40 \\
			\hline
		\end{tabular}}
		